Symplectic forms on fiber bundles and the Chern classes

Abstract

In this thesis we prove two theorems about symplectic fiber bundles (E,π,M,(F,σ)). The first theorem states that there exists a symplectic form on total space E that restricts to induced symplectic forms on the fibers π^−1 (p), if there exists a symplectic form on the base M and there exists a de Rham cohomology class on E that restricts to the de Rham cohomology class of induced symplectic forms on fibers π^−1 (p). The second theorem states that there exists a de Rham cohomology class on E that restricts to the de Rham cohomology class of induced symplectic forms on fibers π^−1 (p), if the first Chern class c_1 (TF) of the tangent bundle of the fiber F is a nonzero multiple of the de Rham cohomology class of the symplectic form σ on F.